As digital equipment becomes popularly used in recent years, analog signal processing is being substituted by digital signal processing. The analog signal processing is performed in continuous time systems, while the digital signal processing is performed in discrete time systems. Therefore, in order to process an analog signal in continuous time systems by means of digital signal processing, there is required a transformation from continuous time systems to discrete time systems.
In general, an object system is expressed by a transfer function defined by a mathematical model. For example, when designing a digital filter or a digital controller for controlling an object, it is necessary to transform the transfer function of an object system that is inherently categorized in continuous time systems to a transfer function in discrete time systems.
In such a so-called discretization method for transforming a transfer function in continuous time systems into a transfer function in discrete time systems, there is a known method called s-z transform. This method has widely been introduced in publications, one of which is “A point of Digital Signal Processing” by Ishida, Yoshihisa et Kamata, Hiroyuki, Sangyo Tosho Publishing Co., Ltd. Basically the s-z transform is a method for transforming the s-plane to the z-plane. In the s-plane, vertical axis is a frequency axis of jω of which range is ±∞ (infinite), while, in the z-plane, a unit circle having a radius of 1 corresponds to a frequency axis, having a finite range.
Accordingly, in the s-z transform, it is necessary to transform a frequency of an infinite interval to a frequency of a finite interval. However, in order to correspond each other one by one, there exist some restrictions. Therefore it is difficult to discretize so as to coincide frequency characteristic, impulse response, step response, etc. completely.
For example, there has been a standard z-transform method in which a transfer function of analog continuous time systems is transformed to a transfer function in discrete time systems with coincident step response. For this reason the method is referred to as the step response invariant method (or impulse response invariant method). In this standard z-transform, as shown in FIG. 22, when the frequency range of the infinite interval in the s-region is matched to a frequency range of the finite interval in the z-region, the frequency axis transformed to the z-region is repeated at certain periods when being extended to the infinite frequency region again. The repetition cycle equals to the Nyquist frequency fn specified by the sampling theorem, which is equal to a half of a sampling frequency fs.
When this folding is made, as shown in FIG. 22, an overlap distortion (or an alias distortion) is produced by the occurrence of the overlap in the amplitude characteristic, causing that the amplitude characteristic in analog continuous time systems cannot be maintained. For this reason, as shown in FIG. 22, the application of this standard z-transform is restricted to a filter that is limited frequency range against a high frequency region.
For example, as illustrated in a frequency characteristic diagram according to the standard z-transform method shown in FIG. 24, in case of transforming a transfer function in continuous time systems in which the amplitude is not sufficiently restricted in the region higher than the Nyquist frequency fn (25 kHz in this example), the characteristics (amplitude and phase) of the discrete time systems transformed by the standard z-transform greatly deviate from the characteristics of continuous time systems in a high frequency region.
In contrast, in the bilinear z-transform, generation of such an alias distortion is prevented. As shown in FIG. 23, the region of infinite length (ωa) in the s-plane is matched to a finite region (p), and then the standard z-transform is carried out against the finite region p. According to this method, an alias distortion is no more produced because the frequency axis having the region of ±∞ in the s-plane is projected to a unit circle on the z-plane when performing the s-z transform. Here, the relation between the angular frequency ωa in the s-region and the angular frequency ωd in the z-region is expressed as follows:ωd=(2/T)·atan(ωa·t/2)  (1)where T is a sampling period in discretization and atan is an abbreviation of arctangent. As can be understood, because a trigonometric function tan θ can take values between ±∞ in the region of −π/2≦θ≦π/2, the alias distortion can be avoided.
Using this bilinear z-transform, accurate discretization can be obtained when poles and zero points of the original transfer function in continuous time systems are located sufficiently lower than the Nyquist frequency fn (a half of the sampling frequency in discrete time systems).
On the other hand, when poles and zero points are located near the Nyquist frequency or higher than the Nyquist frequency, it is often occurred that the characteristics in the high frequency region of the transformed discrete systems deviate greatly from the characteristics in continuous time systems, which causes a problem of such a method.
More specifically, the frequency characteristic diagram according to the bilinear z-transform is shown in FIG. 25. As shown in this figure, when the poles and the zero points of the transfer function in continuous time systems are located near the Nyquist frequency (25 kHz in this example) or higher than the Nyquist frequency, the characteristics (amplitude and phase) in discrete time systems being transformed by the bilinear z-transform (illustrated by the solid lines) greatly deviate from the characteristics in continuous time systems (illustrated by the dotted lines) in a high frequency range. This characteristic deviation in the high frequency range is produced by the transformation in which the characteristics in the high frequency being shifted (which is referred to as ‘warping’) to a low frequency range by tan θ in the above formula (1).
Further, using the warping in the bilinear z-transform is defined by formula (1), there has been proposed a method that only pole and zero point of the transfer function in continuous time systems before the discretization are shifted (frequency shifting) to replace with new transfer function in continuous time systems (which was disclosed, for example, in the official gazette of Japanese Unexamined Patent Publication No. Hei-5-210419, etc.) However, according to this method, it is not possible to transform when the poles and zero points of the transfer function in continuous system are located at higher frequency than the Nyquist frequency because tan θ in the above formula (1) becomes greater than π/2, which produces a negative frequency value after the shift. Therefore this frequency shift method cannot be employed and it is still difficult to make the characteristics coincident.
Accordingly, there has been a problem in the prior art that only an inaccurate transfer function having different characteristics is obtained when a transfer function being designed and identified in continuous time systems is intended to discretize to perform digital processing.
In order to construct such a system according to discrete time systems from a system originally designed in continuous time systems, for example, the following methods have been necessary: In case of performing digital control by performing discretion of compensator of a feedback control system having been designed in continuous systems, there has been employed a method of either substituting an analog control circuit without discretization in respect of poles and zero points located near the Nyquist frequency, or raising a sampling frequency so that the Nyquist frequency is shifted to a higher frequency.
Using the above-mentioned measures, it is possible to obtain the characteristics in discrete time systems being approximate to the characteristics in continuous time systems. However, there is a problem that characteristics of analog circuits vary, in contrast to digital circuits, because of production dispersion or aged deterioration in electronic components. Also, analog circuits become costly when it is intended to use in mass-produced control units. When employing another method mentioned above, high-speed processors are required to raise the sampling frequency. This becomes also costly and disadvantageous considering the use of such processors for mass-produced control units.